Theorem snidg 3400

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
snidg	|- ( A e. V -> A e. { A } )

Proof of Theorem snidg

Step	Hyp	Ref	Expression
1		eqid 2040	. 2 |- A = A
2		elsng 3390	. 2 |- ( A e. V -> ( A e. { A } <-> A = A ) )
3	1, 2	mpbiri 157	1 |- ( A e. V -> A e. { A } )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  snidb  3401  elsn2g  3404  snnzg  3485  snmg  3486  fvunsng  5357  fsnunfv  5363  1stconst  5842  2ndconst  5843  tfr0  5937  tfrlemibxssdm  5941  tfrlemi14d  5947  en1uniel  6284  onunsnss  6355  snon0  6356  fseq1p1m1  8956  elfzomin  9062  bj-sels  10034